OFFSET
0,2
COMMENTS
In Novak's note it is mentioned that if P(z) and Q(z) are the g.f.s for the probabilities of indecomposable and decomposable loops, respectively, that P(z) = 1 - 1/Q(z). This works equally well using loop counts rather than probabilities. The g.f.s may be expressed by the series constructed from the sequences of counts of loops of length 2*n. Q(z) for the 5-d case is the series corresponding to A287317.
To satisfy this g.f. equation, a(0) should be 0, but we give it as 1 since there is one trivial loop of 0 steps, and for consistency with related sequences.
To obtain the probability of returning to the point of origin for the first time after 2*n steps, divide a(n) by the total number of walks of length 2*n in d dimensions: (2*d)^(2*n) = 100^n.
LINKS
Jonathan Novak, PĆ³lya's Random Walk Theorem.
FORMULA
G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^5 dt.
MATHEMATICA
walk5d[n_] :=
Sum[(2 n)!/(i! j! k! l! (n - i - j - k - l)!)^2, {i, 0, n}, {j, 0,
n - i}, {k, 0, n - i - j}, {l, 0, n - i - j - k}]; invertSeq[seq_] :=
CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk5d[n], {n, 0, 15}]]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Shel Kaphan, Mar 09 2023
STATUS
approved